The equation you've provided is \( y = 6 \tan\left(\frac{x}{2}\right) - 3 \). This represents a trigonometric function with a tangent component that is scaled and shifted.

### Key Features:
1. **Amplitude and Vertical Stretch**: The coefficient 6 in front of the tangent function stretches the graph vertically by a factor of 6.
2. **Phase Change**: The argument \( \frac{x}{2} \) indicates a horizontal stretch by a factor of 2. The period of the tangent function, normally \( \pi \), is doubled to \( 2\pi \).
3. **Vertical Shift**: The subtraction of 3 moves the entire graph down by 3 units.

### General Shape:
- The basic shape of the tangent function is maintained, with vertical asymptotes where the tangent is undefined (where \( \frac{x}{2} = \frac{\pi}{2} + n\pi \), for any integer \( n \)).
- The vertical asymptotes occur at \( x = \pi + 2n\pi \) for integer \( n \).
- The function will cross the x-axis where the argument of the tangent is zero, specifically at \( x = 0 \), and its periodic nature will cause it to cross again at intervals of \( 2\pi \).

Would you like a graph of this function or more details on any specific part of it?  

### Related Questions:
1. How would the graph change if the coefficient of \( \tan\left(\frac{x}{2}\right) \) was different?
2. What are the exact locations of the vertical asymptotes?
3. How does the vertical shift affect the maximum and minimum points of the function?
4. What happens to the function as \( x \) approaches the asymptotes?
5. How would you determine the x-intercepts of this function?

### Tip:
When analyzing trigonometric functions, it's helpful to identify any stretches, shifts, or transformations before graphing to understand the behavior of the function more clearly.

Understand the properties and graph of the function y = 6tan(x/2) - 3. Learn about vertical stretching, horizontal stretching, and vertical shifting in trigonometric functions. Suitable for high school students in Grades 10-12 studying advanced trigonometry.

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